# efficiency.py - functions for computing node, edge, and graph efficiency
#
# Copyright 2011, 2012, 2013, 2014, 2015 NetworkX developers
#
# This file is part of NetworkX.
#
# NetworkX is distributed under a BSD license; see LICENSE.txt for more
# information.
"""Provides functions for computing the efficiency of nodes and graphs."""
from __future__ import division

from itertools import permutations

import networkx as nx
from ..utils import not_implemented_for

__all__ = ['efficiency', 'local_efficiency', 'global_efficiency']


@not_implemented_for('directed')
def efficiency(G, u, v):
    """Returns the efficiency of a pair of nodes in a graph.

    The *efficiency* of a pair of nodes is the multiplicative inverse of the
    shortest path distance between the nodes [1]_.

    Parameters
    ----------
    G : :class:`networkx.Graph`
        An undirected graph for which to compute the average local efficiency.
    u, v : node
        Nodes in the graph ``G``.

    Returns
    -------
    float
        Multiplicative inverse of the shortest path distance between the nodes.

    Notes
    -----
    Edge weights are ignored when computing the shortest path distances.

    See also
    --------
    local_efficiency
    global_efficiency

    References
    ----------
    .. [1] Latora, Vito, and Massimo Marchiori.
           "Efficient behavior of small-world networks."
           *Physical Review Letters* 87.19 (2001): 198701.
           <http://dx.doi.org/10.1103/PhysRevLett.87.198701>

    """
    return 1 / nx.shortest_path_length(G, u, v)


@not_implemented_for('directed')
def global_efficiency(G):
    """Returns the average global efficiency of the graph.

    The *efficiency* of a pair of nodes in a graph is the multiplicative
    inverse of the shortest path distance between the nodes. The *average
    global efficiency* of a graph is the average efficiency of all pairs of
    nodes [1]_.

    Parameters
    ----------
    G : :class:`networkx.Graph`
        An undirected graph for which to compute the average global efficiency.

    Returns
    -------
    float
        The average global efficiency of the graph.

    Notes
    -----
    Edge weights are ignored when computing the shortest path distances.

    See also
    --------
    local_efficiency

    References
    ----------
    .. [1] Latora, Vito, and Massimo Marchiori.
           "Efficient behavior of small-world networks."
           *Physical Review Letters* 87.19 (2001): 198701.
           <http://dx.doi.org/10.1103/PhysRevLett.87.198701>

    """
    n = len(G)
    denom = n * (n - 1)
    # TODO This can be made more efficient by computing all pairs shortest
    # path lengths in parallel.
    #
    # TODO This summation can be trivially parallelized.
    return sum(efficiency(G, u, v) for u, v in permutations(G, 2)) / denom


@not_implemented_for('directed')
def local_efficiency(G):
    """Returns the average local efficiency of the graph.

    The *efficiency* of a pair of nodes in a graph is the multiplicative
    inverse of the shortest path distance between the nodes. The *local
    efficiency* of a node in the graph is the average global efficiency of the
    subgraph induced by the neighbors of the node. The *average local
    efficiency* is the average of the local efficiencies of each node [1]_.

    Parameters
    ----------
    G : :class:`networkx.Graph`
        An undirected graph for which to compute the average local efficiency.

    Returns
    -------
    float
        The average local efficiency of the graph.

    Notes
    -----
    Edge weights are ignored when computing the shortest path distances.

    See also
    --------
    global_efficiency

    References
    ----------
    .. [1] Latora, Vito, and Massimo Marchiori.
           "Efficient behavior of small-world networks."
           *Physical Review Letters* 87.19 (2001): 198701.
           <http://dx.doi.org/10.1103/PhysRevLett.87.198701>

    """
    # TODO This summation can be trivially parallelized.
    return sum(global_efficiency(nx.ego_graph(G, v)) for v in G) / len(G)
